In the art of ultrasonic machining as described above, the vibratory energy applied to the tool is transmitted to the cutting front, i.e. the tool-to-workpiece interface, in which the abrasive medium such as diamond, tungsten-carbide, boron-carbide or boron-nitride particles may be present to enhance the cutting action, to effectively work on various materials such as ceramics, calcined or vitreous materials, graphite and so on. According to the prior-art practice, the vibratory energy or oscillations applied the tool and hence transmitted to the cutting front are commonly uniform in nature and also commonly require a preset frequency and amplitude for a given machining operation. Nevertheless, where the workable surface contains a curvature, it has been recognized that there results a considerable surface (finish) irregularity in the ultrasonically machined surface.